So, you have a number of important projects due at work and you wake up one morning to a high fever, headache and congestion. You feel terrible but the company needs you, what should you do? STAY HOME, DON'T WORK! If you're remote, don't log in or check emails until you feel better.
There is nothing noble or inspiring about working while you are ill. It's dumb and selfish. By refusing to take the time you need to get well you are putting others that you come in contact with at risk.
There are many people at the office, grocery store, and other places you frequent that may have weaker immune systems than you. In addition, the company you are working at will survive without you for a couple a days. Hopefully the place isn't so poorly run that one person being absent for a couple days will be its demise.
More importantly, should you, unfortunately, succumb to your illness, which of the following do you think will happen?
If you chose anything but 'C', you are deeply out of touch with reality. Despite what recruiters and hiring managers tell, a company is not a family, its a business that exists to make money. The people who will truly mourn your passing are those family and friends that you have to neglect while working.
As soon as you feel ill, let the appropriate parties at work know you will be taking time off to rest. There is no need to go into vivid details describing your symptions. Doing so is both weird and unecessesary. The company is not entitled to know every detail of what's going on with your body.
Put yourself and your family first always. You only have one life and it should not revolve around one job. On average, people have about 12 different jobs in their lifetime, why kill yourself over something you are probably going to leave in a couple months or years anyway?