Before you even think about using lubricant, have you straightened and smoothed the wires before attempting to insert them? Taped the tips together if necessary? Considered a metal old work box with screw clamps for easy insertion? Deburred any adverse molding marks or seams on the clips? There are 5 or 10 types of single gang old work box that might be legal in your area and some might be vastly easier to install than others.
Examine the part that must pass the squeeze clip for any ridges that can be temporarily taped and examined the box for any hard edges that can be bevelled without hurting the integrity of the box. Can you pre-place a tool like a wire hook to help guide the wires if they're out of finger reach? Depending on the order of how you're trying to insert the cables in the box and insert the box in the wall, you may need to pre bend the cable into a shape that will allow it to move with only linear friction when you pull it into the box.
Lube has a very limited chance of helping you because usually in this situation it is the bend and shape of the wire or an edge or hook creating the problem, rather than simple friction, which lube will help with. Thoroughly examine what you're trying to do mechanically, because excessive forces that would require lube can damage the wire, and the you need to replace the entire length of the wire. Avoid any situation that will tightly or repeatedly bend the wire. Different types of wire casing have different amounts of surface friction and abrasion resistance, so be aware of that. Sharp edges can be bevelled and/or taped over, and the same can work for cloth coatings, that fray, poof and bunch up.
You need to add to your question where you live, whether ground wires are present and the volume of the box you intend to use. Method of calculating necessary box size varies by region. Is it the largest box available? In some cases a box of a different type or size might be much easier to install without cutting more drywall, although that could necessitate a special cover. Fully consider your situation, including cutting and patching wallboard before you risk wrecking your wire and necessitating a lot of additional damage and expense.
Worst case scenario if you damage them is probably ripping drywall all the way back to the panel and to the other branches in order to install new wire. This may necessitate additional inspections and renovations depending on your local code.
If you don't feel confident before you begin, bear in mind the hefty cost of a professional visit is likely much smaller than the cost of a bad error.