わたしのボーイフレンドは日本人とエンジニアです。

わたしのボーイフレンドは日本人とエンジニアです。
Is this sentence correct?

Correct or not depends how many boyfriends you have.

If you have only one boyfriend, you should change the sentence into:

わたしのボーイフレンドは日本人でエンジニアです。
= My boyfriend is Japanese and he’s an engineer.
or
わたしのボーイフレンドは日本人のエンジニアです。
= My boyfriend is a Japanese engineer.

If you have 2 boyfriends and if one is a Japanese and the other is an engineer, the caption sentence is correct.*

This difference is the same difference between:

バナナのケーキをたべた。
= I ate a (piece of) banana cake
and
バナナとケーキをたべた。
= I ate a banana and a (piece of) cake.

I think everybody learns quite early on that verbs/adjectives cannot be connected using the particle と. We need to use the てform instead. Some examples of that are:

昨日（きのう）は買（か）い物（もの）に行（い）って、クリスマスプレゼントを買（か）いました。[2 verbs]
= I went shopping and bought Christmas presents yesterday.

このセーターは10年前に編み始めて、昨日仕上げたものです。[2 verbs in the contact clause]
= This jumper is what I started to knit 10 years ago and completed yesterday.

私（わたし）の母（はは）は優（やさ）しくて、おもしろい人（ひと）です。[2 いadjectives]
= My mum is nice and funny.

このケーキはあまり甘（あま）くなくておいしいです。[2 いadjectives]
= This cake is not too sweet and delicious.

私（わたし）の父（ちち）は元気で活発（かっぱつ）です。[2 なadjectives]
= My father is healthy and active.

この果物（くだもの）は新鮮（しんせん）でおいしいです。[なadjective + いadjective]
= This fruit is fresh and delicious.

The tricky bit is that the same rule applies to nouns when 2 nouns are describing the SAME thing/person.

私（わたし）の息子（むすこ）は学生（がくせい）でサッカー選手（せんしゅ）です。
= My son is a student and soccer player.

このネックレスは母（はは）からのプレゼントで私（わたし）の宝物（たからもの）です。
= This necklace is a present from my mother and (is) my treasure.

The bold で in the above examples are the て form of です/だ. Being a student and a soccer player are the attributes of one person (in this example “my son”), so you cannot use と here. Similarly, a gift from my mother and being a treasure to me are both attributes of the same one necklace, so you need to connect them with で instead of と.

In case of the caption sentence or the examples of a banana cake, however, you can rewrite it using の as well:

わたしのボーイフレンドは日本人のエンジニアです。
= My boyfriend is a Japanese engineer.
バナナのケーキをたべた。
= I ate a (piece of) banana cake

This happens when the first attribute mentioned (a Japanese / banana) is or can be an attribute of the second (an engineer / cake). If 2 attributes you are going to mention have equal importance, probably it’s better to stick to using てform

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