It's a weird one. It feels completely awful to me - I drop games to a local Yoshi who is way, way below my skill level (and I bop him easily with any of my secondaries), but it's possible that I'm somehow just playing the MU wrong, or the local yoshi is just better at the MU than most others, or some other weirdness.
Now that my opinion is properly qualified and hedged, yeah, it's godawful. Yoshi's grab is not slow -- look at the frames on the dash grab. Avoiding that for Bowser is a total nightmare, especially when accounting for the additional threat of the command grab, Yoshi's DJC stuff, and Yoshi's own armor. The way I play Bowser is pretty trade-heavy, and Yoshi is one of a few characters that tend to beat Bowser in trading situations, which makes the MU extremely awkward for me to play.
The eggs aren't really a problem. The problem is that Yoshi out-trades Bowser, has an amazing grab (with amazing followups - notably uair), a solid command grab, and has a far better suite of smashes/kill moves than Bowser. In my opinion, it's just extremely difficult for Bowser to get any traction in the matchup; IMO it's one of his worst along with ROB and Ganon.
It doesn't help that Bowser doesn't really have any way to break Yoshi's doublejump armor, and is one of the worst characters in the game at footstooling. :/