Couple of factors. Flap down reduces landing speed by a few knots. Those few knots reduce control effectiveness, albeit only slightly. This matters when the wind is gusty and/or near the limit of what you can handle as a crosswind.
Flap down also moves the centre of lift (in a spanwise context) more toward the fuselage. Maybe a few inches. The area of the wing with the flap down produces more lift (and drag, but that's beside the point) as the outer wing section. This has the effect of slightly reducing aileron effectiveness.
All this should not matter for a x/w of 6 knots in a 172. If the crosswind was above the demonstrated component, or very gusty up to that component, then it starts to matter. (And then available length/performance considerations need to be made.)
The most important thing is to keep flying the aircraft until it's below minimum flying speed. If the x/w is of variable strength, that means that during the hold-off you are continuously altering the bank angle and required rudder to keep the aligned with the runway and not drifting off to one side or another. At the same time you need to keep flying the aircraft, holding off for a reasonably smooth touchdown at close to normal landing speed. It's a busy time. It's tempting to try and put the aircraft down at the earliest opportunity. Don't. Once the aircraft has touched down, it's tempting to lower the nose, as it's been trained into a lot of pilots that "that's what you do in a crosswind". Don't. Keep the controls positioned to let the nose come down when it's ready. Keep the aileron and rudder inputs as they would need to be for handling the crosswind, as though you were still airborne.
That's how the instructor who kindly corrected my initially poorly learned crosswind technique* taught me, and it's served me rather well since.
My 2p.
(Flare and kick it straight just prior to touchdown, wings level, put the nose down. This technique will bust a nosewheel, sooner or later.)