Most reference I could find only consider European options, but I would like to know whether this also holds for American options in general (with continuous dividend yield and/or discrete dividends)?
This is a good question.
See my answer to a question here
The point is that under Black-Scholes (and also many SV models) not only European prices but also American options prices are homogeneous of degree 1 in strike and spot as the optimal exercise time does not affect the homogeneity property in strike and spot price.
Hence also for American options the dollar gamma is the risk-neutral probability density (where maturity date $T$ is replaced by optimal exercise date $\tau$), which is always positive. So gamma for Americans is always positive.