Can a Non-Segwit Miner mine segwit transactions? I understand that non-segwit nodes interpret segwit txes as Anyone-Can-Spend txes, and they cannot validate txes that spend segwit outputs.

But when an old style miner receives a segwit tx to mine he consideres it invalid, right? (That's because it appears to them as a strange transaction with no input and 1 output.)
So, does it follow that segwit txes can only be mined in a block by upgraded miners?
An old style miner would only receive a stripped transaction, i.e. a transaction from which the witness has been removed. They will consider a segwit transaction to be valid but non-standard because one element remains on the stack when resolving the redeem script.

If a miner included this transaction in their block, the block would appear valid to segwit-incapable nodes, but invalid to any segwit-capable nodes as any segwit input would be missing the required witness and thus would not be properly signed. This would lead to the block being rejected by 98% of the nodes on the network, and thus the segwit-incapable miner to lose their block reward as it would not become part of the heaviest blockchain. Segwit-incapable miners can however contribute blocks that only include non-segwit transactions that build on top of the chain that includes segwit-blocks.

This means that only a miner collecting transactions with a segwit-capable node is able to create a valid block that includes segwit transactions.