Are there masculine and feminine ways to say "Have a good day"?
The question is ambiguous: it might refer to different traditionally gendered ways to say «have a good day», or just gramatically distinct ways to say it. But in any case, either interpretation has the same answer: no.
There are a lot of slightly different ways to wish someone a good day in this conventional manner, but none specifically used by male or female speakers only, or towards male or female hearers. That is, «Que tengas un buen día» and its variants are not traditionally gendered.
Likewise, there are several grammatical variants to this phrase, but they don't include differences in grammatical gender for the speaker or the hearer. «Que tengas un buen día» has only one verb, which is in the second person singular, and the second person singular has no grammatical gender distinction in Spanish. You can find this sentence also in its formal version, with tenga (corresponding to usted) or, if addressed to several people, with tengan (ustedes) or tengáis (vosotros).
You could only find grammatical gender here if you actually included the second person plural informal pronoun vosotros (only in Spain), which is vosotras in the feminine, but that's rather farfetched, mainly because salutations tend to be short and the pronoun is most likely to be dropped in any case.
(This should be a comment but it fits better in the answer box. Pablodf76 is of course absolutely correct in his answer.)
I was curious to see if the answer would be different in Russian, since Ivan is a common name in Russian. I took a look at a list of Russian greetings: http://masterrussian.com/blday_greeting.shtml
Inspired by some gendered polite greetings listed there, here's the closest I could get to a gendered greeting in Spanish:
Mucho gusto en verla / verlo / conocerla / conocerlo. (Nice to see you / meet you.)
Note, "la" and "lo" are formal for "you"; "la" is feminine and "lo" is masculine. If we were to use an informal "you," there wouldn't be any gender distinction.
Of course, the only thing the above sentence has to do with "How are you?" is that they are both standard polite utterances used in greeting processes. So let's get back to the actual question, "How are you?".
It's not unusual to tack something else on after "How are you?" in Spanish for added politeness and human connection:
¿Cómo amaneció Ud., don Silvio?
¿Qué tal le va, Señora?
¿Quiúbole, Eva, qué me cuentas?
The first two, being formal, are explicitly gendered through "don" and "señora," respectively. These terms are similar to "sir" and "ma'am" in English. The last example of the three, being informal, is even less explicitly gendered. The gender of the person receiving the greeting is known and very implicitly expressed, in that "Eva" is a name typically given to a female.
So, if we consider the common functional equivalents of "How are you?", then I would make a very rough guess that half of such utterances do include, in practice, an indication of the gender of the person receiving the greeting. (By "half" I don't mean 50% as opposed to 49%. I mean 50% as opposed to 25%. But even that is just intuition, not based on anything formal.)
I am not aware of any gendered differences in style or choice of expression for this question, with respect to the gender of the greeter (i.e. the person asking "How are you?"). In other words, I think that the choice among the many possible ways of asking "How are you?" in Spanish doesn't depend on the gender of the person asking the question. However, that would make an interesting sociological experiment! (If a sociologist were to study this, it would be good also to probe for differences that depend on the gender of the person receiving the greeting as well.)