Here's some mods and easy things to do, you can read more on them here: Sawyer Mods pt. 1 - Prefilter, tube connector and part 2, new syringes and storage bag.
Articles have more pics and links etc, but this is the basic idea:
Mesh storage bag and 10 ml syringe.
There's the 3 syringes, the one Sawyer gives you, which doesn't really seem to fit anything, the big replacement, which fits perfectly into the tube, and the little one, which can just be pressed onto the tube end for a good fit.
The 10 ml syringe would certainly be better if it were 15 or 20 ml, but it does provide enough pressure for backflushing, and it is totally portable and fits fine with the Squeeze. Takes six fills/plunges to equal one of the big one's.
Note that with the tube / hose connector, the irrigation syringe with the long tip fits right into the tube, and makes backflushing super easy. If I were on a long hike with resupplies, I'd carry the little syringe with me, and put the big one in a bounce box or use it at home, depending on how long I was gone for.
The prefilter is an idea I've come across here as well as other places, it's easy to do, I'd prefer metal mesh, but I couldn't find any, so I used nylon, but it's fine I think.
These are called irrigation syringes, and cannot be found at normal big box pharmacies, those are useless, I went to a real medical supply place, most big cities have them, otherwise you can order online.