If two binary RVs are uniform, they are independent if and only if the probability that they are equal is the same as the probability that they are unequal (so knowing one does not help in guessing the other).
P[X_1 \oplus X_2=X_2\oplus X_3]=P[X_1=X_3] =P[X_1\oplus X_3=0]
which is clearly $1/2$ since $X_1,X_3$ are unbiased and independent.
In general: No.
You have to assume the attacker knows what you are doing. That's Kerckhoffs' principle.
If your random blocks are actually a real One-Time-Pad on top of the block cipher, then it would be stronger - but you could achieve the same without the block cipher. If all your blocks use the same pad, this is true if the length is 1 block, the pad is ...