Any non-SQL Server-process that consumes CPU may cause the entire SQL Server to be completely unusable.
As an example, Windows Update will bring SQL Server to a stop while it is unpacking .Net assemblies.
Any Anti-Virus process, even if it is configured to exclude SQL Server data files and folders may cause SQL Server to become unresponsive while it scans files and while it unpacks updated virus definition files.
If you have any management software installed, such as IBM Tivoli Server Monitor, and it decides to use 100% of one core, you probably won't be able to do anything other than cold shutdown the machine. Don't ask me why I'm bringing that up.
I could go on, but suffice it to say, any server with a single core is simply asking for trouble with performance and manageability. Check that, any computer with a single core should be considered unusable today. Heck, my phone has 8 cores.
While you're at it, convince the virtualization guy to give each SQL Server a reasonable memory reservation that will ensure SQL Server's memory doesn't get paged out of the hosts main memory to disk.
For serious VMware-hosted SQL Servers, you should configure VMware according to the Architecting SQL Server on VMware Best Practices guide.
It really boils down to a single-core VM can only ever run a single thread at any given time. Virtualization is not a silver bullet for that one-core-per-thread situation.
The above points take into account that Windows NT (the 1993 codebase which all modern versions of Windows are built on) utilizes preemptive multi-tasking. Preemptive multi-tasking allows the O/S to enforce multi-tasking by forcing context-switching at regular schedules. Even with preemptive multi-tasking, a server with a single core will still suffer from very poor performance if a single task decides to use as much CPU as possible, since the O/S will give that task as much CPU time as can be spared while still allowing higher-priority tasks to run, forcing CPU to 100%. If SQL Server is running at the time CPU goes to 100%, SQL Server's quantum of CPU will be reduced so much as to become very noticeable. In the above scenarios I outlined, I did not say, the machine (or even SQL Server) will completely "stop" or "die"; that would simply be untrue. However, the machine may become so busy that for all intents-and-purposes the machine becomes unusable. Adding a second core, while certainly not a panacea, reduces the likelihood that the machine will become completely unresponsive. Each additional core means another thread can run to 100% CPU without causing the machine to become unusable.