A hub is a node that has a higher degree than other nodes in the graph. How this translates in practice seems to vary between papers. Some just use an arbitrary threshold either as a fixed number or as a percentage, others choose a threshold based on some statistical argument e.g. outliers in the distribution when assuming a scale-free graph.
As a side note, be careful about what you say about high-degree genes/proteins. Many of them have high degrees simply because they have been more studied (e.g. the historical oncogenes such as the RAS family, SRC, EGFR...) or because they belong to a particular class (e.g. ubiquitin-like families, chaperones) or because they are sticky proteins (some are well documented false-positives such as ferritin and ribosomal proteins for yeast two-hybrid or collagens in AP/MS, see also for example the crapome database for "contaminants" in AP/MS).
Finally, note that the scale-free assumption probably doesn't hold for many biological networks.
Hub genes are highly interconnected genes. There are several methods and approaches for measuring interconnectedness, see for example the cyttoHubba paper for a list of methods.
While some papers just take the top X genes (following some interconnectedness score, or the intersection of several scores) as hub genes, ideally, a minimum requirement should also be required - I've seen often "genes with a connectivity degree of ≥ 8 were defined as hub genes" or "genes with a connectivity degree of ≥ 10 were defined as hub genes".