Does every planet have a solstice and an equinox?

Well, yes, sort of. They would be constant and the same time on a planet with a rotational axis that is perfectly orthogonal (perpendicular) to the orbital plane, so while there would be no SPECIFIC day that applies, the condition would be satisfied albeit permanently.

There is no reason for a particular planet to have such a regular motion that the recurrence of the solstice and equinox would be according to a biannual pattern as on earth, although that would be very likely. The wobble and/or precession of the axis would have to be pretty fast (or the orbit so large) that the orientation of the axis could move enough to matter from cycle to cycle ("year to year") and thus make the arrival of the solstice and equinox conditions also vary by date from year to year.

I suppose we should also recognize that the axis could be parallel to the orbital plane as well, in which case the same sort of condition as perpendicular would apply.

No, only the ones whose axes tilt significantly and whose days are short enough to vary in length over their orbit.

Sort of. The different tilts in their axis would mean wildly different solstice and equinox, however.

Not if they are tidally locked to the Sun like Mercury is.

Yes

They all do

Have you followed the Cassini Mission to Saturn?

The real question you are asking here is: does the direction of the rotation axis of a planet changes as it orbits its host star?

Every planet in our solar system does...

Only two conditions come to mind where there wouldn't:

first is if the orbital plane is exactly identical to a planet's rotational plane... then, it would be an 'equinox' all year long.

second would be if a planet was tidally locked with it's star - then, there wouldn't be a solstice or an equinox.

They don't HAVE to, but most do

It depends on the orientation of the planet's axis. If it is tilted like the Earth's then you will get a solstice and equinox. If the axis is at right angles to the equator of the Sun then all the days are the exact same length.

For the moron who gave this answer a thumbs down I suggest you read the link below

Nope.