Yes, you would be able to open up an individual 401(k) for your business. The catch is that your contribution to that account would be limited to the earnings from that business (after deductible business expenses). If you want to contribute $19k to the 401(k) for your business you'll need to actually earn that much from the business.
These plans have such high limits because you're wearing two hats as a self-employed person: you are the employee and the employer at the same time.
Wearing your employee hat, you can put in 100% of your net earnings up to the standard $19k employee contribution limit, just like you could as an employee with a permissive 401(k) plan.
Past the $19k limit you'll need to put on your employer hat. You're essentially making matching contributions at this point. You can't contribute 100% of earnings for these, it's more like 20%, so to get the full $55k into your solo 401(k) your business will need to pull in a six-figure sum.
Here's a good link about your options as a self-employed person.
