Friend w/o bank account is getting inheritance. Asking if money could be deposited into my checking. Wondering if a deposit of over 300,000 will flag IRS?

Sounds like money laundering, don't do it. 

Anyone can open a bank account so tell your "friend" to open an account and have the "inheritance" deposited into that account.

Ditto.   If your "friend" gets $300,000 there are plenty of  banks who will be very pleased to open a account for your friend.

Friend needs to open their OWN bank account ... with that kind of money any bank should let them open one. 

Once the money is in your account what is going to happen ?   Are you just going to give them it all in cash ?  Movement of large sums of money will trigger bank paperwork and you both need to file gift tax returns.   

Or are you going to dole it out a bit at a time ... what a logistical nightmare and it opens you up to accusations of wrongdoing, theft or fraud.    I highly recommend you do not do this favor for your friend as finances can ruin a friendship fast. 

Sounds like your friend is being scammed or worse ... if you deposit that check your bank account could be at risk.