Just got the JavaJig by Melitta at Walmart for $5.98. This kit "includes: 2 Reusable Cups & 30 Coffee Filters". The two "reusable cups" are 3 parts each, a green outer "cup", a black inner frame and a black lid or cover. The "Coffee Filters" look exactly like miniature paper basket filters (Mr. Coffee type), pleated skirts, flat bottom. Looks like the same paper, thin, open, white. This refillable cup has less capacity for coffee gounds than the other units because the inner frame and the paper filter pleats take up room in an already small container. This setup is somewhat time consuming to clean, reassemble and load, but the paper element prevents any grounds getting in your coffee cup.
The problem with the paper filters breaking is that very fine grind coffee, like Gevalia, will clog the pores of the paper and won't let the water pass thru. Try using a slightly coarser grind. A coarser grind means there is less total surface area of the coffee grounds for the hot water to react with and you will get a slightly weaker brew, which requires slightly more coffee grounds.