Check-calling with the nut flush draw preserves range strength and prevents getting blown off your equity by aggressive raises.
Checking is the preferred play here. This board texture heavily favors the Button's flatting range, which is dense with sets (55, 44), straights (86s, 63s), and two-pairs.
Note: Betting into a range advantage on a low, connected board is risky; checking allows us to realize equity more safely.
Once we bet and face a raise, we have a mandatory call. We have the nut flush draw and an overcard, giving us roughly 55% equity against the Button's raising range.
Checking is correct. The Jack is a relatively neutral card that doesn't change the nuts, and we are still in draw-mode.
Facing a small 1/3 pot bet, we have a close decision between calling and folding. While the price is excellent, our hand is currently just Ace-high.
Note: Calling is a marginal mix, but folding is technically higher EV as we block some of the bluffs we want Villain to have.
The river is a total brick. We have no showdown value and must check. Since Villain checked back, we lose to their entire value range and most of their missed draws.