Is benzocyclobutadiene aromatic or anti-aromatic?
Benzocyclobutadiene has an aromatic benzene ring as well as an anti-aromatic cyclobutadiene ring. This gives it the characteristics of both aromatic and anti-aromatic compounds.
Now, according to Cyclobutadiene and its related compounds, [1,p 180]:

Benzocyclobutadiene, the monobenzo derivative of cyclobutadiene, is intermediate in structure between the unstable parent compound, cyclobutadiene and it stable dibenzo analog biphenylene. Benzocyclobutadiene and its derivatives resemble cyclobutadiene rather than biphenylene in being known as only transient reaction intermediates.

Reading this, we can say that benzocyclobutadiene resembles cyclobutadiene more than biphenylene and so has a dominant anti-aromatic character. This means that like cyclobutadiene, the cyclobutadiene part of the ring undergoes dimerisation to form a more stable compound.
Note: The preview of the book referenced has an entire chapter based on just benzocyclobutadiene. If interested, you can go through it for more information
References:

Cyclobutadiene and Related Compounds; M.P. Cava, M.J Mitchell; Chemistry Department, Wayne State University, Detroit 1967. ISBN 978-01-2164-450-5. [Preview]