I'm looking at a distressed property that has four of six units occupied. Two units are paying only $400/month for 3 bedroom, 1 bath. The other two are not paying anything. The seller is very eccentric to put it kindly. In negotiations he says he has no lease agreements with any of his tenants, not willing to provide any records or other documents.

How would you handle this situation? I'm likely to need to evict the non paying tenants straight away. I'll try cash for keys first then go to court if needed. This is in Landlord friendly Texas so I'm not too concerned with evictions. The other two paying tenants are $300 under market rent even taking into consideration the condition of property. I usually try to honor existing lease agreements but in this case there is nothing in writing. Do I give the paying tenants 30 day notice of rent change to market rent? The remaining two units I plan to fix up and rent at market rent.

Any recommendations?

Knowing nothing else, this seems like a situation where the seller called up some squatters to occupy the home and took in people just to occupy the space well under market value with little/no screening. But if you've seen it and still willing to proceed you would know best if that seems to be the case or not. It just seems like a major red flag to me to not have any leases, so if anything I actually see those tenants as a burden. I would request you get that property 100% vacant and not have to deal with a mess of a situation (possibly legally too). 

Are you looking to buy for cash flow or is this a value add opportunity with rehab involved? No leases will be an issue for immediate cash flow as evicting most likely all tenants will take time and money. 

I've purchased a property where the seller had crammed the apartment complex to 100% occupancy with low quality tenants, he literally took everyone who had enough for security deposit and first month's rent. I've also come very close to another small complex where the seller had allowed (wanted) squatters to live and occupy some units to show occupancy. The property we bought took a long time to get everyone out, touch up the units, and rent with good qualified tenants. The other property I ran as fast as I could from that situation.

@Pete Harper could be an opportunity.  I agree with @Ujwal Velagapudi in asking that it be delivered vacant.

If that is not possible, I would anticipate the downtime and cost of evictions, even without a lease.  Cash for keys is always a great option. You might be able to get them out quicker via any squatting/trespassing laws since there is no agreement in place, but likely formal eviction is needed.

As for trying to keep them, I would general recommend you don't. The feeling I get from reading you post is that a) they won't be able to afford market rents and b) could be a big hinderance getting the 2 vacancies rented.  Of course you will likely need to provide notice of rent increase first, then when they don't pay post notice of eviction and follow laws to get them out.

@Ujwal Velagapudi The Seller is not so devious. Just an eccentric old man who has gotten in over his head. Squatters are most likely old tenants who just stopped paying. 
@Evan Polaski I do not think the seller is capable of getting the old tenants out. If he was able to evict he would have done so already. I failed to mention he is in bankruptcy. He owns a bunch of properties and is selling stuff off to pay creditors. I'm getting rock bottom price so I've factored in carrying costs. The two vacant units look rentable and will carry the mortgage while I get the other four ready.
My main concern is he-said, she-said arguments should I have to drag tenants to court. 

I think most of this depends on your financial position, @Pete Harper . Bank financing will be difficult with 67% economic occupancy and no financials. In fact, you probably won't be able to get any kind of traditional financing. HML may be an option, if you need it.

Ideally, you buy cash, empty everyone out, do all of your renovations, and then re-tenant the property. Once you're up and running go to a bank with delayed financing. Typically, commercial lenders want 1-year seasoning. Keep that in mind.

@Jaysen Medhurst I have bank loan pre approval based on income from other properties. I have a very low debt to equity ratio and solid rental history on other properties. This is our current business model. Buy distressed properties, value add through renovations, raise rents, BRRR. We are just finishing renovations on a 4-plex in similar condition. In this case I signed the two existing tenants up to one year leases at the same rate they were paying. I'm using that income to help renovate the other two. We finished renovations on first unit and rented for $200 more a month. The second unit will be ready in July. The kitchen order has been delayed due to Covid.