It used to stay on/checked, at least until you closed Writer and reopened it. Now it “forgets” whenever the Find and Replace dialog box is closed. There doesn’t seem to be an Options setting for this either. (I only use the Find and Replace dialog when I need to use Regular Expressions.)
There seems to be a setting in Tools -> Options -> Advanced -> Button: Open Expert Configuration -> Search for: IsUseRegularExpression (org.openoffice.Office.common). However it behaves strange and I couldn’t yet figure out how to toggle that setting (It seems to get somehow reset to the current value in some cache/register/whatever of the running program)
There is such a setting, but my LibO V 7.0.3 remembers the last choice if I call the F&R dialog (Ctrl+H), and writes the last state to the user profile when I close the application. From there it is also read again when I “go Expert Configuration” next time.