Are conjugate priors required when performing Gibbs sampling?
You don't require conjugate priors for Gibbs sampling.
What you need to be able to do is produce samples from the full conditional distributions.
Conjugate priors generally make that easier, but easier is not the same as necessary.
In particular situations there may be any of a variety of ways of producing samples from the full conditionals without a conjugate prior.
For example, one approach that is sometimes usable is via accept-reject; if you can bound the ratio of an approximation you can sample from the prior x likelihood, that may be doable (e.g. if you can bound the ratio of the density of some mixture of conjugate priors to the actual prior).
Of course, possible doesn't necessarily imply its always the best available solution to the issue.