Your course of action really depends on how much work and effort you can put into the expired domain. If you're barely able to maintain and optimize your current site, you probably want to just 301 redirect the old site (note: see my amended comment above about the link value not likely to be passed). If, however, you're more creative and have some time on your hands, you can try your hand at crafting a microsite. If you really know your stuff and are experienced at making money off various websites, you'd probably do well with the third option.
Also, I currently have two other domains that are registered with Squarespace, and I have their annual payments set up automatically so I never have to manually pay (and so nobody else gets that window of opportunity to take the domain away from me). Would it be possible that the owner of this domain might just have a similar system in place and that I might be wasting my time?
The bottom line is, drop domains deliver traffic. How come? When people register a drop domain and build a website on that domain, they often attract backlinks. They also can update their website and generate a community or a brand. Whatever the case is, there are four things that happen that results in traffic for those websites. They create a backlink footprint, which drives direct traffic. This backlink footprint also produces SEO benefits. The more these website owners promote their website, the more goodwill they build up in their target niches and online communities. Finally, the more they manage their online blog, website or information site, they gain some sort of presence on social media.
ExpiredDomains.net gathers all the information you need to find good Expired Domains that are Pending Delete and you can Backorder. Depending on the domain extension you can search through thousands of domains every day before they get released to the public and pick what you like. ExpiredDomains.net currently supports 473 TLDs. From the classic gTLDs like .com, .net, .org to Droplists for ccTLDs you can only find here and now we even support some of the best new gTLDs like .xyz and .club.