I went on an all meat diet for 10 days. The 10th day I noticed that I had a bruise on the inside of my leg. Could it be that my veins blew up? ?

Doubtful. Unless you are consuming a huge amount of fish oil (even then it is still unlikely to happen in 10 days), I'd doubt your diet has any effect on the bruises. Have you been exercising more vigorously during the same period.?
Leak: Yes, Blewup No. Veins walls are very thin & ones close to surface more easily injured without noticing the trauma if distracted. Any likelihood of relationship to eating protein+fat foods? Very, very unlikely. Hi saturated fat, moderate protein, low to zero carb = most healthy. Healthy blood glucose about 3/5th level teaspoon, or less, in one's entire blood volume. https://goo.gl/t2VMMy http://goo.gl/cypKOp.
Low Vit.C/flavonoids. Your vein didn't blow up but this is almost certainly due to a deficiency of Vit. C and bioflavonoids as there's NONE of those in meat.Our bodies can't store Vit. C so we need it EVERY DAY. C is essential for the health and maintenance of collagen, the connective tissue in the wall of blood vessels, so low C results in unhealthy vessels and increased bruising.If you try this again take a multivit.