Olá Ksenia, tudo bem? Na minha opinião você está correta em pensar que a frase deveria ser "'[...] tem que calçar as suas botas.' Ou seja, no plural. No entanto, é muito comum ver as pessoas aqui no Brasil dizendo sapato/bota - no singular. Parabéns pela atenção a esse ponto!

[2/2] 'ele me fez cócega' or 'minha costa tá doendo'; in the case of 'óculos', we actually never say 'óculo', but we might treat it like the words of the first category, saying something as 'cadê meu óculos?' for just one pair of glasses. I believe the reason for this is that we feel them as just one entity (after all, we have only one 'costas' and 'óculos' are a single object). That is NOT recommended, though; so, except in a daily, laid-back setting, avoid that (especially in the case of 'costas', since 'costa', in the singular, is another thing: a coast), saying instead 'ele me fez cócegas', 'minhas costas estão doendo', and 'cadê meus óculos?' (here you need the context to know if this refers to a single pair of glasses or more than one). The reason is usually historical or logical: 'costa' used to refer to a rib, so 'costas' would be a group of them (the back); 'óculo' would be a single glass lens (as in a monocle), so two of them make up 'óculos'; and 'cócega' would be a single tickle, but you never feel just one when being tickled, do you? So... 'cócegas'. 4) Words with normal singular and plural forms: bota(s); sapato(s); luva(s). There is no mystery here: 'a bota', 'o sapato', and 'a luva' refer to a single item for just one hand or foot; 'as botas', 'os sapatos', and 'as luvas' refer to more than one item, usually a pair for both hands or feet. Now we get onto your actual question: =) yes, it should be 'calçar as suas botas'. But, as Claudiano pointed out, informally it is very common (at least here in Brazil) to say something like 'calce a bota', 'que tênis/sapato bonito!', 'coloca a tua luva', etc.; not because we think of them as a single entity (as with a 'calça')—they are clearly two separate items—, but because, in most contexts, it's obvious that putting one of them on also implies putting the other on, so you can treat the pair using the singular word. Hope this has helped you and I've made myself clear! Let me know if you have any questions. =)

[1/2] Good catch, Ksenia. =) Let's take advantage of your question and explain the bigger picture. Although we don't have so much confusion as with uncountable nouns in English (you know, 'a pair of glasses/pants', 'a piece of advice/news', etc.), we do have a few words that might be confusing when it comes to their singular or plural usages. So, let's divide them into four categories: 1) Words with the same singular and plural forms: ônibus; lápis; tênis. As we know, normally in Portuguese the letter S (or '-is' or '-es', for that matter) is used to make the plural of words. In this case, the singular already has a final S, so the plural is identical to it. The only thing that distinguishes them, in this case, is the articles, numerals, etc., and/or the context: 'o ônibus' (singular), 'os ônibus' (plural); 'um ônibus' (singular), 'três ônibus' (plural). 2) Words that can be used either in the singular or the plural to refer to just one item: cueca(s); calça(s); short(s). Traditionally, they should be used in the plural, usually for the same reason as in English: they are items made up of two 'parts'. But, unlike English, Portuguese has evolved and today it is acceptable to use them in the singular when referring to a single item; I'd say that, at least in Brazil, it's even way more common to do that rather than use the historical plural versions. So, 'a cueca', 'a calça', and 'o short' refer to a single item, while 'as cuecas', 'as calças', and 'os shorts', in theory, can either refer to a single item or more than one, but you can bet your farm that, at least in Brazil, they will refer to more than one item for the majority of the time (especially 'cuecas'). 3) Words that are used only in the plural: cócegas; óculos; costas. Those words refer to a single entity, but they should only be used in the plural: 'as cócegas', 'os óculos', 'as costas'. BUT, in informal Brazilian Portuguese, it is actually sort of common for them to be treated in the singular, as in (+)

For some reason, italki messed up my answers, changing their order, so I had to edit them just to put them in the right order, but I didn't actually change any word. :)