What exactly is your problem?

There much stuff here.

I think you are trying to parametrize the rectifiable curve.

You divide the line integral into 3 parts:

1)From (0,0,0) to (1,1,0)

2)From (1,1,0) to (1,1,1)

3)From (1,1,1) to (0,0,0)

For #1 use the curve,

<x,y,z>=<t,t,0> for 0<=t<= 1.

For #2 use the curve,

<x,y,z>=<1,1,t> for 0<=t<=1.

For #3 use the curve,

<x,y,z>=<t,t,t> for 0<=t<=1.

The evaluation I leave to thee.