Can the user get the private key from the encrypted and the plaintext data or does it make it easier for an attacker?
No, that would make no sense.
If a person could get the private key from encrypted data, that would be akin to locking something and leaving the key in the lock. Likewise, the decrypted message is the same - bit for bit - as the message before any encryption took place. Therefore, the message has no relation to the private key at all.
A few more details
There are several attacks on RSA that could allow an attacker to recover the private key, such as the low exponent attack. These attacks are mitigated by using proper parameters for RSA. This should be seen as "if RSA is used incorrectly, an attacker can calculate the private key". It is not a "feature" of the encryption algorithm, nor is it something that is easily done with some openssl command.
What you asking is known as Known-Plaintext Attack or in short KPA. When RSA has properly implemented the answer is NO!.
Actually, the attackers don't need to capture and access the plaintext in public-key cryptography. In public-key cryptography, the encryption is free, since you know the public key of the target, that is the pair (e,n), then you can encrypt as many pairs as you want. Formally we call this access to Encryption Oracle is free.
it must be infeasible for a computationally bounded adversary to derive significant information about a message (plaintext) when given only its ciphertext and the corresponding public encryption key.
The semantic security is equal to Ind-CPA ( indistinguishability under chosen-plaintext attack) and this is a stronger assumption than KPA.
In a chosen-plaintext attack the adversary can (possibly adaptively) ask for the ciphertexts of arbitrary plaintext messages
Of course, there were attacks on the RSA other than the factoring over the years. This article covers many of them;