I know the down syndrome - abortion thing on twitter, honestly if I had a fetus with downs I will abort it myself as well. (And I'm liberal...)
Rightwingers and liberals/leftist opposed him on that one, from different positions. Rightwingers opposed him because they consider the fetus a human life and it reminds them of eugenics. Liberals and leftists opposed him because of saying that its immoral to not abort. Someone said that saying that its immoral to not abort is the same as to say that abortion in that case is immoral.
Don't know about eugenics, although I doubt Dawkins support Hitler-style eugenics.
I don't think he was referring to forced sterilization. I think he was referring to designer babies, selecting for high intelligence/iq etc. Some in the hbd movement might support voluntary sterilization of people with low iq and criminals, I don't know whether Dawkins would support that. If he did support something like that publicly, it could cause the twitter storm of the decade.