Do phonetic symbols have names in English?

Solution 1:

Sometimes. A few symbols have specific names that are normally used. Many have one or two names that are often used, but you can also describe the sound or the letter.

Some tips:

• IPA symbols that are also English letters can take their English names (e.g. ⟨s⟩ /ɛs/ "ess"). However, you may instead choose to describe the represented phone, when the English pronunciation differs greatly from the IPA one (e.g. ⟨j⟩ perhaps "palatal approximant" rather than "jay").
• ⟨ə⟩ is the schwa, as already pointed out. However, "schwa" is sometimes used more generally to refer to any roughly mid-central vowel, regardless of rounding and precise position.
• ⟨ɛ⟩ and ⟨ɔ⟩ are open E and open O, which can equally describe the letters and their phonetic values.
• Letters that look like Greek letters may be called by the English names for the corresponding Greek letters: ⟨ɛ⟩ epsilon, ⟨θ⟩ theta, ⟨ʊ⟩ upsilon, ⟨β⟩ beta, ⟨ɣ⟩ gamma, ⟨χ⟩ chi, ⟨ɸ⟩ phi.
• Those that represent English phonemes that don't have letters in the English alphabet are named by analogy (nasals and voiceless fricatives /F/ are pronounced /ɛF/, like ⟨m⟩, ⟨n⟩, ⟨s⟩, ⟨f⟩; voiced fricatives are pronounced /Fi/, like ⟨z⟩ (AmE), ⟨v⟩). These are ⟨ʃ⟩ "esh", ⟨ŋ⟩ "eng". Exception: ⟨ð⟩ is apparently called "edh" /ɛð/ (also spelled "eth") and ⟨ʒ⟩ "ezh" even though you would expect *"thee" and *"zhee".
• Some have conventional names based on their shapes: ⟨ɤ⟩ ram's horns.
• Some are always named using a description of their phonetic value: ⟨ʔ⟩ glottal stop.
• Diacritics and attachments often have names (themselves either traditional like "diaeresis" or based on their IPA meanings), which gives names for letters with those attachments. For example: ⟨ç⟩ C with cedilla, ⟨pʰ⟩ aspirated P, ....
• Otherwise, there's probably no special name, so just describe either the letter (e.g. ⟨ɰ⟩ "turned m with leg") or the sound (e.g. "voiceless alveolar trill"), whichever is clearer or more concise.

Aside: /ʃ/ "esh" isn't actually an integral sign (they're different Unicode code points), though people will likely still understand you since it accurately describes the shape.

Solution 2:

The Unicode standard has names for phonetic symbols. Start with the Wikipedia article on phonetic symbols in Unicode. Since the Unicode standard is an accepted international standard that also specifies glyphs for the symbols, you would be justified in using it in a professional context.