When you're working freelance, there is no such thing as a sick day.
You don't have a job that you can call in sick to. It's only you, the work in front of you, and the deadline. That's it. How you manage the three is totally up to you.
When accepting a job on a short deadline, what you're saying to the client is that you will do the job come what may. As a freelancer, the only thing you have is your reputation. You are only as good as your current job. The last job may have gotten you the current gig, but if you stumble and fall on that current gig, you're harming your reputation. That stumble can occur when you're sick.
The inability to call in sick is terrible. This is because you never know when being ill will strike. Or some sort of accident or emergency. (We use these words for a reason.) This is why, if I'm the editor doing project management and my client has a deadline for things, I do everything in my power to build in a buffer for every step.
A buffer is generally this: if you need it in seven days, you tell the people you're working with that you need it in five. This gives you a two day buffer before you really need it. The bigger the buffer, the more leeway you give yourself if something unforeseen happens. (Here's a tip: as a client, it's always best if you can under-promise and over-deliver. If you have a deadline of five days and you know you can do it in three, say it's five but do it in three. Don't be lazy and do it in the actual five that you have. There may be other people down the line who are waiting for you to finish the work.)
However, if you need the “extra” time because you're sick, you've already got the buffer built in if you've managed your time right.
Manage your time right.
Do as much extra work as you can, as early as you can. You may need that buffer somewhere down the road.