I'm hoping you get a variety of responses, because this is a tough one.
Both your husband and Jen are participating in a sort of subversive/thrilling "one who got away" scenario.
They are not friends. They are nostalgic exes.
I doubt she wants to be with him. But she loves knowing he still "can't get you out of my mind." And she makes sure of that by these tiny, private intrusions "happy birthday," "I'd like that." She steers clear of making romantic overtures because their dynamic is that *she* is in the "power position," *she* is "the one who got away," *she* is the one being pined for. She gives him just enough to stay in that role, which must also provide some kind of either thrill or security blanket for her, or else, like most exes of 20 years, there would either be no contact or actual adult old-friends interactions (such as a friendly birthday wish on the public wall; the private messages reinforce a "special" connection even when their content is "harmless.")
As for your husband, I have direct experience with someone who does this. I have been in both the "wife" and "Jen" positions (with this same person). Here is my take: Your husband uses his connection with Jen to escape the emotional vulnerability and possibly terror of full commitment. He does not see himself as an unfaithful type, and probably experiences absolutely no hypocrisy when he talks about his awful cheating ex.
"Jen," although an actual real person, is actually largely symbolic at this point. I doubt he wants to be with her, either. What he does want is the reassurance that there is this alternate life he could have had. He likes hinting that they could have a drink (which never seems to work out, right?) I doubt he has anything more than a hazy idea of what would then transpire, because, actually, for both of them, really spending time together might kill this mutual nostalgia thing. It is actually almost kept alive by how sporadic it is, and by the fear, on both of their sides, that it could end at any time. So they intermittently ping one another just to make sure the other person is still playing.
If they did get together, and this may have been the dynamic the few times they ran into each other for real, it would be all wistful eye contact and heavy sighs and a kiss on the cheek "take good care of yourself," and one of them would say, "I'll always wonder what if…" and then later, when they got home, one of them would FB PM something like "I must have temporarily lost my mind. Go be sweet to your wife."
The problem for you is that he is carrying a torch and she is blowing some oxygen on it whenever it flickers.
But his torch has very little to do with her as a real person at this point.
Which means, imo, is that, paradoxically, Jen is not the issue. Your husband's use of nostalgia to escape the present and the life he has actually chosen is a much deeper issue.
It is for that reason that, contrary to prevailing SI advice, I don't think that confronting, asking him to go NC, etc. will really address what's going on. Because it is *the absence and can't-have* that she represents that he is addicted to.
It is almost as if, in your case, if you picked up on the next time he, on the flimsiest of pretexts, evokes her name, you said, oh, right, Jen, doesn't she live near here? Let's all go out for drinks next week since she seems to be important to you and I'd like to get to know her. In other words, if you made her into a real, normal person and in all innocence (ha!) trampled all over her gauzy shrine in his mind
at least something would change. If you all got together, the long-lost-lover fantasy would be overwritten with a real experience of you as wife and her as old friend.
This would take immense restraint on your part, as, neither before or during this event would you "confront" or tip your hand, or even give the slightest hint that you know they are in secret contact.
There are several possible results: it could simply burst the bubble entirely. Or they might exchange more private messages in the aftermath that would give you a clearer sense of how invested each is in continuing. Or one or both of them might figure out you know the score, and that might spoil the game and end it.
But the character issue within your husband that wants to have a secret space from you (I always think of men who do this as casting their wives in the "mommy" role, and they are hiding candy from mommy), that romanticizes an escape hatch, is and will be an issue regardless of Jen.
I actually suggest IC (therapy) for you around how this makes you feel and what actions you can take. In a healthy marriage, with a healthy partner, you'd be able to say, "Listen, you mention Jen a lot. She clearly was very important to you. I'm wondering if you sometimes fantasize about what your life would have been like with her." And then that would open up a space for actual communication, and then once Jen-as-symbol has been sort of "outed" within the marriage bond, the mirage will fade.
But what I suspect is that the whole, much deeper problem here, is that you can't have such a conversation in your marriage, because he is not capable of it and values his secrets like a kid with a private fort in the backyard. That he will become irritated and belligerent and want you to stop trampling on his romantic fantasy.
In IC you might be able to talk about how you want to approach this and if you want to accept that part of who your husband is involves juvenile secrets and external ego strokes.
I know this was long, but I just want to validate that this is a real, and significant problem. My sense is that if you confronted him with your evidence and gave a the-marriage-or-your-ex ultimatum, if would backfire and he would feel like you were taking away his favorite toy, and then you would be the big bad mommy and he would use that to reach out to her. I could be wrong, but this is not harmless, he is using it to escape commitment, and it has got to stop, but since it is about *longing and not having* to begin with, what it means for it to stop is a deep internal maturity and boundaries on the part of your husband.