Can a gene continue on from one chromosome to another separate chromosome? For example, can part of an eye colour gene be on one chromosome and the rest of it be on the next chromosome?
Usually not (as described in the other answer, non-contiguous pieces of DNA are not considered as a part of a single gene). However there are some cases of trans-splicing in which the spliceosome can splice together RNAs arising from different genomic locations. But these are rare and do not have a direct effect on distinct phenotypes such as eye colour, to be classified as genes. Not much progress has happened in this aspect of transcriptomics, actually.
No, a gene is confined to a single chromosome. A gene is characterized by a start sequence in the DNA (among many other things), which indicates that the nucleotides are to be read and transcribed to RNA. This stretch of DNA in the gene is contiguous, although some parts may be skipped in transcription. Because different chromosomes are physically distinct, the cellular machinery would require two start sequences, which really means there are just two separate genes. Instead of finding one eye color gene on multiple chromosomes, you instead find multiple eye color genes on multiple chromosomes.
Gene is usually defined as piece of DNA that produces certain product, be it RNA or polypeptide molecule. In both cases RNA is produced as intermediate or final product. Nature of RNA production from DNA (aka transcription) is such that it requires double-stranded DNA as substrate, that is a continuous polymer of nucleotides.
So, only parts of chromosomes (which are really long linear polymers) can be used as substrate or template for RNA production.
However, regulatory elements, which are extremely important, can be anywhere, as long as they are able to function. Since those elements are just special sequences of DNA that interact with transcription factors and other elements of DNA->RNA machinery, they can be on different chromosome than gene of interest.