How big of a deal is this? (Broken seal)

You're not protecting anything by smearing grease all over the outside of it. The part that needs grease is inside the boot.

You are, however, providing a great adhesive for dirt and grit to get picked up onto. It's just gonna be a bigger mess than it was before. Best to clean it all off now... even the grease that leaked out of the boot.

That part is the drop link. Connects the anti-roll bar to the wheel, through the strut body. It's simple to replace (two bolts) and cheap if you just want to buy a new one...

 

These appear to be the upper end of the links between the stablizer bar and the mount on the strut cylinder.

I would first comment that this method of applying grease would/will do little or nothing to lubricate this joint The moving parts are on the inside of the rubber boot. These are lubricated and sealed for the life of the joint, with no grease zerks or fittings. If there is any movement in this ball type joint, it will be evident by movement which may only be detected by pushing up and down on the stablizer bar end with a jack or on a hoist with a pry bar. It there is movement, then the link assembly will have to be replaced (non-rebuildable)

Now for the boot that is now off/not seated. I would first clean all that grease off and check inside the seal to see if there is any dirt. If it looks clean, simply try to snap it back over the ball link shoulder that retains it. If there is dirt, I would recomment just flush, just the seal boot with aresal parts cleaner to get all the diry grease out, mininizing soaking the ball joint it's self to prevent washing the grease out and forcing dirt farther into the join. Wipe things down and apply enough grease to cover the ball joint movement area. No need to completly file the boot which might cause it to pop off again. Don't put grease on the rubber and metal area where the seal snaps over the shoulder since that might also lead to it coming off again.

If you find play in the link joint and opt to replace it, You can do it yourself by rasing the car to work under it and with the car on a jack stand, move your jack under the torsion bar end and move it up until there is no pressure on these two ball joint ends. Loosen the nuts using an allen wrench to hold the stud bolt. If you don't have a puller, when the nut is just flush with the end of the stud, tap it with a soft faces hammer. (if you don't have this, put a soft metal object between the hammer and the nut/stud end to prevent unnecessary to the these thread (you might need them if the new one did'nt come with nuts.)

Definately a DIY with a jack, jack stand and small tools.

My opinion FWIW!

Steve