No. An adversary who knows a ciphertext and one of its plaintext blocks p can trivially find the corresponding xored-plaintext x. Since block ciphers with fixed keys are bijections, if there is
a previous plaintext block then x has at least as much entropy as that previous plaintext block.
(In particular, x can very easily be different from your other first blocks of data.)
However, the adversary would have seen the raw-block-cipher encryption of x
(it's the ciphertext block corresponding to p), and so can tell whether or not a
given ciphertext has x as its first block of data. Furthermore, if they can get x to
be one of your first blocks of data, then they can proceed as in the BEAST attack.
On the other hand, your approach would work if your first blocks of data are
each chosen in a way that does not use any of previously-produced ciphertexts.
(Sketch: By induction, each ciphertext block will be computationally unpredictable,
so the probability of the xored-plaintexts repeating will be negligible.)