Most Helpful Guy
Most players won't admit to the girl that they're players; it'd kill their game. Player or not, though, him making eyes and flirting with other girls while you two are out is entirely unacceptable. I'll admit that I also have low self-esteem and crave validation, but the thought of flirting with other girls when I'm out with a girl has never even crossed my mind, so I don't buy that as an excuse.
I think your instinct is worth following; it definitely shows up as a red flag to me.