Yeah I'll kick the hornets nest: you should not be making egg jokes about strangers, especially to their face. These kinds of jokes are potentially damaging to closeted trans people, because you don't know someone's situation. Maybe they already know they're trans, but are in an unsafe situation. Maybe they just aren't ready to address it. Maybe they'll start doubting if they actually want to transition or they only think they want to because a bunch of people told them they do. And it's especially bad if, say, you're making a transfem egg joke to a trans man who isn't open about his transness, which could lead to serious dysphoria (and vice versa for trans women). Not to mention how terrible it is to do this to nonbinary people. Do not joke about a nonbinary person being a binary trans egg, it's extremely disrespectful and transphobic to do so. And it's hard not to notice how implying someone is an egg only extends towards binary genders. Imagine someone is doubting their gender, and everyone is telling them they're obviously a trans man egg. They might end up forcing themself into that binary because they think that's what everyone wants.
All of this isn't to say you can't kindly and respectfully say "the things you've been saying sound very similar to the trans experience, I think it's possible you may be trans, but ofc it's up to you to decide whether or not to look into it". I very much encourage that, especially if the person is your friend and seems to be struggling. The thing that's really bad is making jokes at the possible closeted trans person's expense, or thinking of "cracking an egg" as something to brag about and not something to help someone through because you care about them
