A continuación el código de:

Login.jsx

1import React from "react";
2import Axios from "axios";
3
4const authentication = (e) => {
5 e.preventDefault();
6
7 const form = e.target;
8 const data = {
9   email: form.email.value,
10   password: form.password.value,
11 };
12
13 Axios.post(`${process.env.REACT_APP_API_USER}/login`, data)
14   .then((response) => {
15     localStorage.setItem("token", response.data.token);
16     window.location = "/";
17   })
18   .catch((error) => console.log(error));
19};
20
21const Login = () => {
22 return (
23   <div className="ed-grid">
24     <div className="l-block"></div>
25     <div className="m-to-center m-60 lg-30">
26       <h1 className="center">Iniciar sesión</h1>
27       <form onSubmit={authentication.bind()}>
28         <div className="form__item">
29           <label htmlFor="email">
30             Correo electrónico
31             <input
32               type="email"
33               id="email"
34               placeholder="Ingrese su e-mail"
35               required
36               name="email"
37             />
38           </label>
39         </div>
40         <div className="form__item">
41           <label htmlFor="password">
42             Contraseña
43             <input
44               type="password"
45               id="password"
46               placeholder="Ingrese su contraseña"
47               name="password"
48             />
49           </label>
50         </div>
51         <div className="form_item">
52           <input
53             type="submit"
54             value="Iniciar sesión"
55             className="button full"
56             required
57           />
58         </div>
59       </form>
60     </div>
61   </div>
62 );
63};
64
65export default Login;

Register.jsx

1import React from "react";
2import Axios from "axios";
3
4const registration = (e) => {
5 e.preventDefault();
6 const form = e.target;
7
8 const data = {
9   email: form.email.value,
10   name: form.fullname.value,
11   password: form.password.value,
12 };
13
14 Axios.post(`${process.env.REACT_APP_API_USER}/signup`, data)
15   .then(() => {
16     alert("Usuario creado");
17     window.location = "/login";
18   })
19   .catch(() => alert("Error al crear el usuario"));
20};
21
22const Register = () => {
23 return (
24   <div className="ed-grid">
25     <div className="l-block"></div>
26     <div className="m-to-center m-60 lg-30">
27       <h1 className="center">Crear cuenta</h1>
28       <form onSubmit={registration.bind()}>
29         <div className="form__item">
30           <label htmlFor="fullname">
31             Nombre completo
32             <input
33               type="text"
34               id="fullname"
35               placeholder="Ingrese su nombre"
36               required
37               name="fullname"
38             />
39           </label>
40         </div>
41         <div className="form__item">
42           <label htmlFor="email">
43             Correo electrónico
44             <input
45               type="email"
46               id="email"
47               placeholder="Ingrese su e-mail"
48               required
49               name="email"
50             />
51           </label>
52         </div>
53         <div className="form__item">
54           <label htmlFor="password">
55             Contraseña
56             <input
57               type="password"
58               id="password"
59               placeholder="Ingrese su contraseña"
60               name="password"
61             />
62           </label>
63         </div>
64         <div className="form_item">
65           <input
66             type="submit"
67             value="Iniciar sesión"
68             className="button full"
69             required
70           />
71         </div>
72       </form>
73     </div>
74   </div>
75 );
76};
77
78export default Register;

.env.local

REACT_APP_API_URL =  http://localhost:3003

REACT_APP_API_USER = https://api-edteam.alejogs4.now.sh

Para que se leída la variable de entorno local, deben parar la ejecución de la app en la terminal y volver a mandarla a producción (con el comando yarn start).