What fabric should I buy for both quilt and hive cover and how big should I cut it?

I bought wood to make some modified Warre hives. (They are about an inch wider -- 14" outer dimensions, and shorter, and the top bars are more narrow.) I'd like to eventually make a lot of hives, because I'd like to make a living from beekeeping. I'm afraid I need to cut the fabric about 18"x18", though if I can cut it to say 15"x15", maybe I could finish more hives with less fabric.

I have a 15% off coupon from Hancock fabrics that I could use to buy 100% jute burlap 36" wide, $2.79 per yard. (I don't know the shipping.) It expires on the 10th, so I need to decide quickly. This means I could get 12 pieces of fabric and finish 6 hives per 3 yards. If I could cut 15"x15" I could get 7 hives per 3 yards.
That is $7.11+shipping and about $1.19 per hive.

The other option I have found is to use canvas painters drop cloth. I found one at lowes.com (again, not including shipping):
Blue Hawk 9-ft x 6-ft 8 Oz. Canvas Drop Cloth $10.98 This would be 12 hives at 18"x18" or 14 hives at 15"x15". That is $.92 per hive.

The drop cloth is cheaper, but I don't know if it is good enough.

I would say any coarse weave cloth would work fine. Just make sure it's not treated with any chemicals.

I have burlap, can I use that for the quiltbox?

I think when you are concerned with a $.27 cent difference in hive costs, you can't really afford to become a beekeeper.

I've used burlap to make a quilt box before and it worked very well.
I filled it 3" or so deep with dried shredded leaves, and the ten frame double deep colony thrived in the sub-zero upstate New York winter.

Most of the leaves stayed quite dry whne I stuck my hand into them in mid winter - only the top 1/4" or so where they contacted the cold air was damp, and that evaporated of fast enough that the dampness never went deeper.

If i still lived in the Lower Columbia Basin, I'd make sure that whatever fill I used was pretty loose and allowed good air flow.
With winters as warm as it is there (unless you are much higher than Elkhorn Mountain) there isn't need for a lot of insulation, though some is helpful.
(We call the mid-thirties in winter a heat wave here.)

i used heavy landscape cloth from lowe's, it did not stretch at all.

Coffee shops will frequently give you the perfect burlap. They buy the beans in burlap bags.

I also used landscape cloth. cheap, easy