Would a bit of grease on overhead door cables or bearings cause the springs to break?
TLDR: The spring failed naturally due to old age or manufacturing defect.  Not your fault. As an aside, learn to properly grease garage doors.

Oh boy.  The bull-oney here is thick.  Let's start with Springs vs Door Motion. 

The Spring is a counterweight. THAT IS ALL.

We're not even talking about lifters yet.  Just the door, with no external forces to move it. Capische?

The door is heavy. You could not hand-lift it.  You should not rely on an active mechanism to sheer-muscle-lift that door.  Why not? First if the lifter breaks,  the door guillotines and everyone under it is dead. Problems start out as minor (and reveal by stiff motion) so you want to detect that early, not brute force it, breaking the door further until guillotine. So muscle-lifting a door is Right Out.  

The trick with counterweighting a door is as it rolls upward, it weighs less and less, because part of it is going horizontal.  That is pretty linear.  Springs are also "pretty linear", so they're a natural.  

The point of counterweighting is that the door weighs effectively zero in all positions and will stay where you put it, not wanting to go up more than down in any position.  

The counterweight interacts ONLY with the door.  If the door moves, the counterweight instantly tracks with it. No interference!  If the counterweight slackens, catastrophe follows.  Which brings us to the First Rule Of Garage Doors: 

Opening forces act ONLY on the door, never the counterweight.  


  We had four large 14x20' tall doors. The door and spring were conventional, but some genius put a geared lifter on the spring shaft. Worked going up. But going down, you were slackening the spring, and hoping gravity would pull the door down to keep the cable taut.  Springs are instant.  They follow the door motion no matter where it goes and how fast.  Doors, not so much, especially when horizontal and gravity isn't working on them. The cable did not remain taut, came off the spool, guillotined... And that's why you never do that. 




So.  Back to your problem.  Assuming your installer did not make the above mistake...   It does not matter how the door moves.  The spring counterweight should be able to keep up with it no matter what. Therefore, anything that affects door motion - like grease in the tracks - has no bearing on the performance of the spring. 

The spring failed all by itself, in its own time, simply because it was old.   Final answer. 

Why are you greasing door tracks?

Grease is not a magical "make things move that should" elixir.  Actually, making things not stick together is a complex problem that you can't just throw one chemical at.  And garage doors are a case in point of this.

Grease, being sticky, is a magnet for dirt and dust. Those are abrasives, so the net effect of greasing is the opposite - you might as well have used spray sandpaper.  Grease makes sense inside areas sealed from the environment, like roller bearings. 



Now what about door tracks? Remember how a door functions. The vertical part of the track, the track is basically a guide.  The door is hung by the counterweight cables, and has basically no horizontal motion at all, except for wind loads. So on a still day, garage door rollers are doing zero work (in the vertical) and certainly don't need any grease. 

But, the horizontal part of the door tracks?  When the door is horizontal, all its weight rests on the rollers, and that's where they earn their pay.  (However if your door is sticky when it's barely open, that's not the problem since it isn't horizontal).  Effectively, the horizontal track is a roadway and the rollers are the bicycle wheels.  Now let me ask you something.  To make your bike roll freely, do you grease the road?  Of course not!  You make sure your bike wheels are top top.  You clean the road to keep debris out of it, because debris slows you down. 

The same here with garage door rollers.  Make sure the rollers are tip-tip.  Don't lubricate them with grease, because dirt magnet.  Rollers are cheap and if they seem to be anything but excellent, just replace them.  As for the horizontal track, make sure it is spic-and-span.  Particularly remove any grease and the accumulated dust/dirt/sawdust that it attracted, that got up there.  



There is one spot that really does need lubrication, but grease really will not do.  You notice the garage door's track is on a very slight angle, so the  door slides slightly  toward the building as it drops.  At bottom, it makes a tight seal to the building.

If the door is sticky at the first few inches of opening, look there.  However, grease is right out, because it'll get on your clothing. Go to a tractor supplier and get a quart of Paint-On Graphite Lubricant and a few acid brushes.  This stuff looks, walks and quacks like dark grey paint, but is nicely slick.  It's oil based, so be prepared to throw the brushes away. 

The graphite is heavy, and really loves to settle to the bottom leaving useless resin at the top. It's like cookie dough on the bottom and water on the top, and it takes 20 minutes to hand stir it to lump-free.  So ask the hardware store to give it a good long run in their paint shaker upside down. (Buy quarts; the gallon is too heavy and would break their paint shaker). 

To keep it from separating while you paint, pour out 1/4" of depth into a paint cup or old soup can, at that depth it will stir as you wet your brush. 

Rollers are cheap and I don't sweat just replacing them, but graphite paint is one thing I will use on rollers with open bearings. A light touch will suffice, don't drown it.