My company is based in Hong Kong, and I'm also not a US resident. My company is doing an online service matching students around the globe. US students will also be the target, and I would like to hire US students as interns/part time to do online marketing work, remotely. In this case, do I need to register and set up a US company (I'm trying to avoid it if possible as I would like to focus on the business until it can take off. My company is still at very early stage)?

Can I hire US interns/contractors by non-US company?
Answers
No, you don't need to be a US company.
Be advised there are a plethora of US and State laws on what is and isn't an independent contractor, an "intern", etc.
Wayne Spivak's comment is correct, but to expand on it just a bit, the challenge for you is if you don't properly set up the independent contractor arrangement and the people end up classed as "employees" under either state law in the state where the person lives or under Federal law, the tax consequences for you could be a disaster.
I recommend staying away from the term "intern". There are a whole different sets of rules associated with that position. If the person is a paid intern, that typically makes the person an employee, which would mean having to register for payroll tax withholding in every state in which you have an intern, as well as registering with the US government. If it is an unpaid intern, you may have a different problem in that wage and hour laws in some places make it difficult to use an unpaid intern for work for which a company would normally pay a regular employee. Stick with treating everyone as an "independent contractor", but as Wayne stated, make sure you completely understand how wage and hour and payroll tax laws in every location treat classify independent contractors vs employees.
Also, if the people you retain as independent contractors doing remote marketing are selling your products to people inside the USA, you need to have someone look at the laws determining whether or not you could be classified as "doing business" in the USA or any state. If your company would be classified as "doing business", that could make you subject to both income tax and sales & use tax.
Before you retain anyone in the USA and start marketing your products there, it seems that a bit more research needs to be done of the laws in the areas in which you want to contract with someone for services and where & how you plan to do the marketing and sales. Someone such as Wayne or another person engaged in fractional CFO services & consulting for small businesses could help. Alternatively, if your audit firm in Hong Kong is a member of an international accountancy network, they could refer you to one of their USA affiliates.
Thought of one other possibility after I signed off from the previous comment. It could be feasible to arrange with an interim staffing agency or PEO to contract with them for the person you want to retain to do work for you. Some agencies will act as the employer of record for people you've found who you want to have working for you.
I'm retired living in Thailand now, which is closer to you, time wise. Message me through my profile if you want to discuss possibilities with someone who has experience working with small and medium sized HK companies doing business in the USA, as well as with US based SMEs.