Is it obligatory to urge Muslims to deal with a company owned by a Muslim instead of a company owned by a Christian in a costly, non-essential public service, or is this considered unethical because it might cause losses to the Christian company?

All praise is due to Allah, and prayers and peace be upon the Messenger of Allah, his family, and companions. To proceed:

It is not obligatory to call upon Muslims to deal with a Muslim company over a Christian one. However, it is religiously encouraged, even if it leads to the loss of a Christian company, because it is an act of benefiting a Muslim brother. The disbeliever often spends his money on the interests of disbelievers. Nevertheless, dealing with disbelievers in buying and selling is permissible as long as the transaction itself is not forbidden or does not involve assisting in something forbidden.