In Hückel's rule, can n be any integer?
No, you do not have to have as many &#x3c0; electrons as p-orbitals or any type of direct relationship. As long as you have a continuous, circular arrangement of p-orbitals and an electron count $4n+2$ (for monocycles), you can expect your molecule to be aromatic.


Pyrrole, furan and related molecules are aromatic. They have 6 &#x3c0; electrons like benzene, but only five p-orbitals.
The cyclobutadienyl dianion has been synthesised as a dilithium salt. The authors claim evidence for aromaticity.[1]
The dianion of cyclooctatetraene has also been synthesised and examined by ESR as early as the 1960’s. The authors state:


  The assumption that both the dianion and the radical-anion are planar […] is consistend with [their equation] within experimental error.


Thus it can be assumed that $\ce{cot^2-}$ is also an aromatic system.[2]
Disubstituted dications of cyclooctatetraene have been synthesised a decade later by Olah et al. The published proton NMR data implies that the protons are more deshielded than a simple triene moiety would imply. The summary the authors give states the following:


  The $\ce{^{1}H}$ and $\ce{^{13}C}$ NMR spectroscopic characterization of the reported new cyclooctatetraene dications indicates these ions possess those characteristics expected of a $\ce{C8}$ 6 &#x3c0; aromatic system and hence further extends the evidence for the great predictive utility of the $4n+2$ Hückel rule.


$\ce{cot^2+}$ is thus also an aromatic system.[3]


These last two bullet points must be the ones you are looking for. The cyclooctatetraene dianion, a cyclic $\ce{C8}$ species with 10 &#x3c0; electrons ($n=2$) is aromatic, as is also the cyclooctatetraene dication, the same $\ce{C8}$ species albeit with 6 &#x3c0; electrons ($n=1$).

For most systems, aromaticity is either already given (as they have e.g. six &#x3c0; electrons in their neutral state; compare benzene etc.) or they are only one electron away (cyclopentadienyl, cycloheptatrienyl or tropylium). The way these molecules obtain aromaticity is relatively fixed since greater charges destabilise molecules.

Only for those systems starting with $4n$ &#x3c0; electrons such as cyclooctatetraene, which are antiaromatic by the Hückel definition, can there be a choice. Both the dication and the dianion are generally unfavourable, but they are similarly unfavourable thus the system can be pushed into either direction.

It is highly unlikely that a tetraanion or tetracation of benzene exist outside of a $4~\mathrm{K}$ matrix, much like it is very unlikely for a cyclopentadienyl-trication or a cycloheptatrienyl-trianion to exist outside of similar conditions due to their great charge density.



References:

[1]: A. Sekiguchi, T. Matsuo, H. Watanabe, J. Am. Chem. Soc. 2000, 122, 5652. DOI: 10.1021/ja0004175.

[2]: H. L. Strauss, T. J. Katz, G. K. Fraenkel, J. Am. Chem. Soc. 1963, 85, 2360. DOI: 10.1021/ja00899a004.

[3]: G. A. Olah, J. S. Staral, G. Liang, L. A. Paquette, W. P. Melega, M. J. Carmody J. Am. Chem. Soc. 1977, 99, 3349. DOI: 10.1021/ja00452a027.