The author of that article is 29 years old. Having a boyfriend IS embarrassing at that age, because that's exactly the point in life when the words "my boyfriend" or "my girlfriend" start sounding juvenile coming out of your own mouth.
The phrase signals that you're still unsettled, still caught up in the frivolities of dating that many of your peers - and more importantly, the older and higher-ranking professionals who you're hoping to *make* your peers - have already left behind. You find yourself in the middle of more conversations about spouses and families, and become increasingly aware of the fact that you have absolutely nothing to add to them. When you say "my boyfriend" or "my girlfriend" you immediately feel like a kid at the adults' table.