Just like aromatic compounds, anti aromatic compounds have several resonance structures and some of them are also conjugated ( 1,3-cyclobutene).
Why is it that just having 4n electrons makes them unstable?
In conjugated-pi ring systems of N atoms, you typically have a single low energy orbital and then paired (degenerate) energy levels until you have N orbitals total. This has to do with the number of nodes in the conjugated pi system, and the number of ways you can lay them out with N atoms.
Now, when you add in electrons you add two electrons per orbital, so your 4n+2 electrons fill 2n+1 orbitals -- the single low energy orbital and then n complete sets of degenerate pairs.
If you only have 4n electrons, you can fill the single low energy orbital, then 2(n-1) complete sets of degenerate pairs ... but with two "extra" electrons which aren't paired up. This allows those two electrons to "spread" out between the pair of equal-energy orbitals. Those unfilled/partially filled orbitals mean that the anti-aromatic compound is quite reactive.
The other issue is that "stability" is not an absolute property. Stability is always in reference to some other state. The typical anti-aromatic systems you'll encounter tend to have those frontier orbitals as non-bonding or anti-bonding orbitals, meaning the electrons in the orbitals aren't really contributing to the molecular stability. Generally, if you build the energy level diagrams for alternate systems (e.g. a bent ring where you don't get localization) you'll find that - while the lowest energy orbital isn't as low energy as in the ring-conjugated system - the sum of the energies across all orbitals is indeed lower. And it's the total energy of the system which is optimized. (For example, in cyclobutadiene the ring-conjugated form would have two electrons in a low-energy bonding orbital and two in non-bonding orbitals. In contrast, linear butadiene bonding orbitals are not as low energy as for the ring-conjugated cyclo form, but there's two of them. And the sum of the energies from two linear bonding orbitals is lower than what you get from a single cyclo bonding orbital.)
(Final note: this is all high-level generalization. There's more sophisticated levels of quantum mechanical theory where things get more complex.)