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Does his min-raise say "made flush", "made top pair with flush draw", or "made bluff"?

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Given your description of him, I'd lean towards the latter two. But also given that I probably go for a check-raise on the flop since he will probably bet and I want to blow him off the hand before something scary comes off.

Whether or not you want to reraise all-in now or wait to push a non-Q or -club turn depends on other factors. I lean towards the push because, well, I think I'm ahead and I don't want to lose what has become a pretty sizable pot-- especially since he has folded to your reraises in the past. If it's earlier in the tournament, I lean towards keeping the pot small in close situations, but your overbet has eliminated that possibility.

If he calls I expect to see top pair and a club (probably not a very good one). I'd expect any hand that hit stronger than yours to raise preflop (except Q9 or two clubs).