Do you WANT a relationship? Because if you do, then lying and saying you have a boyfriend is working against your own best interests.
If you DON'T want a boyfriend/relationship, then, sure, say what you like.
There's nothing embarrassing about being single. You can say "my last relationship didn't work out and I'm a little shy so it usually takes me a bit longer to find another one." That's completely truthful and something almost anyone can understand (and something that FAR more people then you probably imagine can relate to).