Not necessarily. Socialism is centralized resource control, but it can coexist beside private businesses. Think about single-payer health insurance. Several countries use a mixed system where the government just buys from private insurance companies on behalf of the citizens. The system is a mix of the two economic models.
Besides, a democratic government is essentially a business that's owned by all the people in the country. Each citizen has an equal share in the government. One could argue that a private business that doesn't allow equal representation of its workers is against freedom and pro-authoritarianism as a result, since the business owner(s) are profiting off the laborers, creating a miniature monarchy/oligarchy. Yes, people could find a new job at a new business, the same way they can find a new residence in a new country, but when all you have are dictatorships, there's not any pro-freedom places to go to.